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Though not always consistent, the children's game show Legends of the Hidden Temple had an end game that can become unwinnable depending on certain situations. First, there were the Pendants of Life, needed to get past three Temple Guards that will yank a contestant out of the temple during the end game if they don't have a full one, and which are rewarded in a Golden Snitch-type 1-1-2 three game system; one half pendant for the first two games, a full one for the last. Because of this, it's possible to make it to the end game with only 1 and a half or even a singular Pendant (though in the case of the former, the show gives the contestants the chance to find the other half-Pendant inside the temple), and depending on where the Temple Guards are hiding and which doors in the temple are locked, it's very possible (and has happened several times in the show's run) to be forced to encounter all three Temple Guards with only one pendant, a definite no-win situation.
If the team is doing well enough it can also be a no-lose situation. If the team is doing well enough and gets BOTH pendants, then they cannot lose unless they were to run out of time.
Although, there did seem to be one HUGE design flaw that no one ever seemed to notice or fix. After a while, it appears that they designed, built, and tested the temple on adults. When the typically shorter kids ran through the temple, there were a few places where they struggled. Jester's Court was pretty terrible, because many kids were unable to stretch themselves enough to hit the buttons at once.
There were also a few pretty terrible cases where the game was unwinnable by mistake through a Game-Breaking Bug. There were instances in which teams wound up taking an unnecessary amount of time because it was not recognizing the button pressing. Perhaps the worst case was in one game where a temple door accidentally closed and re-locked by someone who just passed through it. When their partner came in, they were stuck and baffled because they saw their partner pass through that door.
In the first season of The Amazing Race, three teams were essentially eliminated on leg nine, since the next leg had a strict Hours of Operation time limit that made it impossible for the two teams who technically did survive to ever catch up to the lead pack. Over the next four legs, the 3rd and 4th place teams were arriving at the Pit Stops over twelve hours behind the top two teams, meaning they were actually arriving after the leading teams had already started the next leg. This meant that by the last episode of the season, they were doing tasks that the other teams had completed in the previous episode, making their continuing to race merely a formality. Subsequent seasons added deliberate equalizers, points at which teams are forced to be evened up with each other, to go along with the looser "bunching points" that caused many of the problems near the end of Season 1.
...and on All-Stars, Bill & Joe, the same team who got screwed by the equalizers in Season 1, were caught in ANOTHER Unwinnable situation. The course designers had accidentally scheduled leg 6 to coincide with a religious holiday in Africa, which screwed up the airports and once again put them over 12 hours behind. They did manage to somewhat catch up to the pack by leg 8, but by then they were slapped with the Marked for Elimination penalty to overcome on a very short leg containing an Intersection (which made it impossible to finish more than 30 minutes ahead of the other teams) and a Fast Forward (which prevented them from finishing first), which is pretty much impossible to pull off.
Similarly; accidents have caused the game to become unwinnable for individual players. Such as players accidentally losing their passports or money, or even injury.
Strictly Come Dancing once had to cancel the elimination in the three-way semi-final when the third-placed couple could not escape it even had they won the public vote. The votes were carried forward to the final, and future series would have five couples in the semi-final.
In-universe example: On the sitcom Newhart, George Utley invents a board game called "Handyman" that becomes a local smash hit. You gain 3 points every time you land on a scoring space, and there is no other way to gain or lose points. You win when you score exactly one million points. Three does not divide evenly into one million.